IGCSE Edexcel Engineering: Formula & Theorem Quick Reference Guide | IGCSE Edexcel 工程：公式定理速查手册

📚 IGCSE Edexcel Engineering: Formula & Theorem Quick Reference Guide | IGCSE Edexcel 工程：公式定理速查手册

This quick reference guide covers the essential formulas and theorems required for the IGCSE Edexcel Engineering specification. Each section presents key relationships with clear explanations, helping you revise core concepts in mechanics, materials, electronics, and thermal systems efficiently.

本速查手册涵盖 IGCSE Edexcel 工程课程的核心公式与定理。每个小节都提供了关键关系式并附有清晰的解释，帮助你高效复习力学、材料、电子和热力系统的基本概念。

1. Stress and Strain | 应力与应变

Stress (σ) is the internal force per unit area within a material when an external load is applied.

应力（σ）是外部载荷施加时材料内部单位面积上的内力。

σ = F / A

Strain (ε) measures the deformation of a material relative to its original dimension. For tensile or compressive strain, it is the change in length divided by the original length.

应变（ε）衡量材料相对于原始尺寸的变形程度。对于拉伸或压缩应变，它是长度变化量除以原始长度。

ε = ΔL / L₀

Where F is the applied force, A is the original cross‑sectional area, ΔL is the change in length, and L₀ is the original length.

其中 F 是施加的力，A 是原始横截面积，ΔL 是长度变化量，L₀ 是原始长度。

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2. Young’s Modulus | 杨氏模量

Young’s modulus (E) is a measure of the stiffness of a material. It is the ratio of tensile stress to tensile strain within the elastic limit.

杨氏模量（E）是衡量材料刚度的指标，是弹性极限内拉伸应力与拉伸应变之比。

E = σ / ε = (F L₀) / (A ΔL)

The unit of Young’s modulus is pascal (Pa). A higher E value indicates a stiffer material that deforms less under load.

杨氏模量的单位是帕斯卡（Pa）。E 值越高，材料越硬，在载荷下变形越小。

For many metals, Young’s modulus remains constant until the yield point. Engineering design uses E to predict elastic deflections in beams and shafts.

对于许多金属，杨氏模量在屈服点之前保持恒定。工程设计利用 E 来预测梁和轴的弹性挠度。

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3. Moments and Levers | 力矩与杠杆

A moment (M) is the turning effect of a force about a pivot. It depends on the magnitude of the force and the perpendicular distance from the pivot to the line of action of the force.

力矩（M）是力绕支点产生的转动效应，取决于力的大小和支点到力作用线的垂直距离。

M = F × d

The principle of moments states that for a system in equilibrium, the sum of the clockwise moments equals the sum of the anticlockwise moments.

力矩原理指出，对于处于平衡状态的系统，顺时针力矩之和等于逆时针力矩之和。

Levers are simple machines classified into three types depending on the relative positions of load, effort and pivot. The law of the lever can be expressed as:

杠杆是一种简单机械，根据负载、动力和支点的相对位置可分为三类。杠杆定律可表示为：

effort × effort distance = load × load distance

This relationship allows a small effort to lift a large load if the effort distance is sufficiently large.

如果动力臂足够长，该关系式使得较小的动力就能举起较大的负载。

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4. Mechanical Advantage and Efficiency | 机械利益与效率

Mechanical advantage (MA) is the factor by which a machine multiplies the input force. It is the ratio of load to effort.

机械利益（MA）是机械放大输入力的倍数，即负载与动力的比值。

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）是相同时间内动力移动的距离与负载移动的距离之比。

VR = distance moved by effort / distance moved by load

Efficiency (η) measures how well a machine converts input work into useful output work, always less than 100% due to friction.

效率（η）衡量机械将输入功转化为有用输出功的程度，由于摩擦总是小于 100%。

η = (MA / VR) × 100%

In all real machines, MA < VR because some energy is lost to heat and sound.

在所有真实机械中，MA < VR，因为部分能量以热和声音的形式损失了。

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5. Gears and Gear Ratios | 齿轮与传动比

Gears transmit rotary motion and torque between shafts. The gear ratio is determined by the number of teeth on each gear.

齿轮在轴之间传递旋转运动和扭矩。传动比由每个齿轮的齿数决定。

gear ratio = number of teeth on driven gear / number of teeth on driving gear

A ratio greater than 1 indicates a speed reduction and torque increase; a ratio less than 1 gives a speed increase and torque decrease.

传动比大于 1 表示减速增扭；小于 1 则增速减扭。

The output speed can be found from:

输出转速可由下式求得：

output speed = input speed × (teeth on driver / teeth on driven)

Torque is inversely proportional to speed. Neglecting losses, power is conserved:

扭矩与转速成反比。忽略损耗时，功率守恒：

input torque × input speed = output torque × output speed

When designing gear trains, the overall ratio is the product of the individual pair ratios.

设计齿轮组时，总传动比为各对传动比的乘积。

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6. Ohm’s Law and Circuits | 欧姆定律与电路

Ohm’s law states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points, provided the temperature remains constant.

欧姆定律指出，在温度保持不变的条件下，通过导体两点之间的电流（I）与这两点之间的电压（V）成正比。

V = I × R

Resistors in series have the same current. The total resistance is the sum of the individual resistances.

串联电阻的电流相同，总电阻等于各电阻之和。

R_total = R₁ + R₂ + R₃ + …

Resistors in parallel share the same voltage. The reciprocal of the total resistance is the sum of the reciprocals of each resistance.

并联电阻的电压相同，总电阻的倒数等于各电阻倒数之和。

1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …

These formulas are fundamental for analysing electric and electronic systems in engineering.

这些公式是分析工程电气和电子系统的基础。

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7. Electrical Power and Energy | 电功率与能量

Electrical power (P) is the rate at which electrical energy is transferred by a circuit. It is given by the product of voltage and current.

电功率（P）是电路传递电能的速率，等于电压与电流的乘积。

P = V × I

Using Ohm’s law, alternative forms of the power equation can be derived for resistive loads:

利用欧姆定律，对于电阻性负载可推导出功率的另外两种形式：

P = I² × R    and    P = V² / R

Electrical energy (E) consumed over a period of time is the power multiplied by the time.

一段时间内消耗的电能（E）等于功率乘以时间。

E = P × t

The standard unit of energy is the joule (J), but for larger quantities kilowatt‑hours (kWh) are often used in engineering contexts.

能量的标准单位是焦耳（J），但在工程中常用千瓦时（kWh）表示较大能量。

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8. Density and Thermal Properties | 密度与热性质

Density (ρ) is mass per unit volume. It is a material property that influences component weight and buoyancy.

密度（ρ）是单位体积的质量，是影响零件重量和浮力的材料特性。

ρ = m / V

When the temperature of a solid or liquid changes, it expands or contracts. Linear thermal expansion is governed by:

当固体或液体的温度变化时，会发生膨胀或收缩。线性热膨胀由下式控制：

ΔL = α L₀ ΔT

where α is the coefficient of linear expansion, L₀ is the original length, and ΔT is the temperature change.

其中 α 是线膨胀系数，L₀ 是原始长度，ΔT 是温度变化量。

The heat energy required to raise the temperature of a substance without a change of state is calculated using specific heat capacity (c):

在物态不变的情况下，提高物质温度所需的热量用比热容（c）计算：

Q = m c ΔT

During a change of state, latent heat (L) is involved:

在状态变化时涉及潜热（L）：

Q = m L

These thermal principles are important when selecting materials for applications like engine components or heat exchangers.

在选择发动机零件或热交换器等应用的材料时，这些热学原理非常重要。

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9. Fluid Pressure and Continuity | 流体压力与连续性

Pressure (p) in a fluid is defined as the normal force per unit area. It is a scalar quantity.

流体中的压力（p）定义为单位面积上作用的法向力，是个标量。

p = F / A

For an incompressible fluid flowing through a pipe of varying cross‑section, the principle of continuity states that the mass flow rate remains constant. For a constant density fluid, this simplifies to:

对于流经变截面管道的不可压缩流体，连续性原理指出质量流量保持不变。对于密度恒定的流体，简化为：

A₁ v₁ = A₂ v₂

where A₁ and v₁ are the cross‑sectional area and velocity at point 1, and A₂ and v₂ are the corresponding values at point 2.

其中 A₁ 和 v₁ 是点 1 处的横截面积和流速，A₂ 和 v₂ 是点 2 处的对应值。

Hydraulic systems use the principle that pressure is transmitted equally throughout a confined fluid. If two pistons of different areas are connected, the force multiplication can be found from:

液压系统利用封闭流体中等压传递的原理。若连接两个不同面积的活塞，力放大倍数可由下式求得：

F₂ / F₁ = A₂ / A₁

This is the basis of hydraulic jacks, brakes, and presses used in engineering.

这是工程中使用的液压千斤顶、制动器和压力机的基础。

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10. Work, Energy and Power (Mechanics) | 功、能与功率（力学）

Work done (W) by a constant force is the product of the force and the distance moved in the direction of the force.

恒力所做的功（W）等于力与在力方向上移动距离的乘积。

W = F × d

Gravitational potential energy (Eₚ) gained by an object lifted through a height h is:

物体被举升高度 h 所获得的引力势能（Eₚ）为：

Eₚ = m g h

Kinetic energy (Eₖ) of a moving object depends on its mass and speed:

运动物体的动能（Eₖ）取决于其质量和速度：

Eₖ = ½ m v²

Power (P) is the rate of doing work. It can be expressed as work per unit time or, for constant velocity motion, as the product of force and velocity.

功率（P）是做功的速率，可表示为单位时间做的功，或在匀速运动时表示为力与速度的乘积。

P = W / t    and    P = F × v

The principle of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. Engineering systems must account for energy losses due to friction, air resistance, and electrical resistance.

能量守恒原理指出能量既不会凭空产生也不会消失，只能从一种形式转化为另一种形式。工程系统必须考虑摩擦、空气阻力和电阻等造成的能量损失。

These mechanical energy relationships are crucial when designing lifting mechanisms, vehicles, and power transmission devices.

在设计起重机构、车辆和动力传输装置时，这些力学能量关系至关重要。

Published by TutorHao | Engineering Revision Series | aleveler.com

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