IGCSE WJEC Engineering: Quick Reference Handbook of Formulas and Theorems | IGCSE WJEC 工程：公式定理速查手册

📚 IGCSE WJEC Engineering: Quick Reference Handbook of Formulas and Theorems | IGCSE WJEC 工程：公式定理速查手册

Engineering at IGCSE level draws together principles from physics, materials science and mechanics, making a solid grasp of core formulas essential for both exam success and practical problem-solving. This handbook consolidates the key equations and theorems you need to know for the WJEC Engineering specification, presented in a clear bilingual format that supports revision and quick reference.

IGCSE 阶段的工程学科融合了物理、材料科学与力学的基本原理，牢固掌握核心公式对考试成功和解决实际工程问题都至关重要。本手册整理了 WJEC 工程规范中需要掌握的关键方程和定理，以清晰的双语形式呈现，便于复习和快速查阅。

1. Forces and Motion | 力与运动

Newton’s second law describes how the resultant force acting on an object causes it to accelerate. It is expressed by the fundamental relation that links force, mass and acceleration.

牛顿第二定律描述了作用在物体上的合力如何使其加速，通过力、质量与加速度的基本关系式表达。

F = m × a

where F is the resultant force in newtons (N), m is the mass in kilograms (kg) and a is the acceleration in metres per second squared (m/s²). This vector equation means that acceleration always occurs in the direction of the net force.

式中 F 为合力，单位牛顿 (N)；m 为质量，单位千克 (kg)；a 为加速度，单位米每二次方秒 (m/s²)。这个矢量方程意味着加速度的方向总与合力的方向相同。

Weight is the force exerted on a mass by gravity. It is a specific application of F = m × a where a is replaced by the acceleration due to gravity, g.

重量是重力施加在物体上的力，是 F = m × a 的一个特例，其中加速度 a 被替换为重力加速度 g。

W = m × g

On Earth, g is approximately 9.8 m/s² (often taken as 10 m/s² for estimation). Remember that weight is measured in newtons, not kilograms.

在地球上，g 约为 9.8 m/s²（估算时常取 10 m/s²）。注意重量单位为牛顿，而非千克。

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2. Moments and Equilibrium | 力矩与平衡

A moment is the turning effect of a force about a pivot. It depends on both the size of the force and the perpendicular distance from the pivot to the line of action of the force.

力矩是力绕支点产生的转动效应，取决于力的大小以及支点到力作用线的垂直距离。

M = F × d

where M is the moment in newton-metres (Nm), F is the force in newtons (N) and d is the perpendicular distance in metres (m). The principle of moments states that for a system in rotational equilibrium, the sum of clockwise moments about any pivot equals the sum of anticlockwise moments.

式中 M 为力矩，单位牛·米 (Nm)；F 为力 (N)；d 为垂直距离 (m)。力矩原理指出，对于处于转动平衡的系统，绕任意支点的顺时针力矩之和等于逆时针力矩之和。

∑ M_clockwise = ∑ M_{anticlockwise}

This principle is widely used in analysing levers, beams and simple machines, and is central to many engineering structures.

该原理广泛应用于分析杠杆、梁和简单机械，是许多工程结构的核心。

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3. Work, Energy and Power | 功、能与功率

Work is done when a force moves an object through a distance in the direction of the force. It represents a transfer of energy.

当一个力使物体沿力的方向移动一段距离时，就做了功，它代表能量的转移。

W = F × s

Here W is work in joules (J), F is the force in newtons (N) and s is the distance moved in metres (m). If the force is applied at an angle, only the component parallel to the displacement does work.

式中 W 为功，单位焦耳 (J)；F 为力 (N)；s 为移动距离 (m)。若力呈一定角度，则只有平行于位移的分量做功。

Power measures the rate at which work is done or energy is transferred. It is a crucial quantity in engineering when selecting motors and engines.

功率衡量做功或能量传递的快慢，是工程中选择电机和发动机时的关键参数。

P = W / t or P = E / t

P is power in watts (W), W is work (J), E is energy (J) and t is time (s). One watt equals one joule per second. For mechanical systems, power can also be calculated as the product of force and velocity: P = F × v.

P 为功率，单位瓦特 (W)；W 为功 (J)；E 为能量 (J)；t 为时间 (s)。1瓦特等于1焦耳每秒。在机械系统中，功率也可用力和速度的乘积计算：P = F × v。

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4. Stress and Strain | 应力与应变

When a material is subjected to an external force, it experiences stress. Engineering stress is the applied force divided by the original cross-sectional area.

当材料受到外力作用时，会产生应力。工程应力是施加的力除以原始横截面积。

σ = F / A

σ (sigma) is normal stress in pascals (Pa), where 1 Pa = 1 N/m²; F is the axial force; A is the cross-sectional area. Stress can be tensile or compressive.

σ（西格玛）为正应力，单位帕斯卡 (Pa)，1 Pa = 1 N/m²；F 为轴向力；A 为横截面积。应力可以是拉伸或压缩应力。

Strain is a measure of deformation, defined as the change in length divided by the original length.

应变是变形程度的度量，定义为长度变化量除以原始长度。

ε = ΔL / L₀

ε (epsilon) has no units; ΔL is the change in length; L₀ is the original length. For many materials, the ratio of stress to strain is constant within the elastic limit, known as Young’s modulus.

ε（伊普西龙）无量纲；ΔL 为长度变化量；L₀ 为原始长度。对许多材料而言，在弹性极限内应力与应变之比为常数，称为杨氏模量。

E = σ / ε

Young’s modulus E is measured in pascals (Pa) and indicates a material’s stiffness. A typical value for mild steel is around 2.1 × 10¹¹ Pa.

杨氏模量 E 的单位为帕斯卡 (Pa)，反映材料的刚性。低碳钢的典型值约为 2.1 × 10¹¹ Pa。

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5. Electricity and Circuits | 电与电路

Ohm’s law is the fundamental rule linking voltage, current and resistance in a conductor held at constant temperature.

欧姆定律是联系恒温导体中电压、电流和电阻的基本规律。

V = I × R

V is potential difference in volts (V), I is current in amperes (A) and R is resistance in ohms (Ω). This linear relationship applies to ohmic conductors only.

V 为电势差，单位伏特 (V)；I 为电流，单位安培 (A)；R 为电阻，单位欧姆 (Ω)。这种线性关系仅适用于欧姆导体。

Electrical power can be expressed in several ways by combining Ohm’s law with the basic power equation.

电功率可以通过欧姆定律与基本功率方程结合，以几种方式表示。

P = I × V

P = I² × R

P = V² / R

These formulas are useful for determining power dissipation, selecting fuses and designing circuits. Energy transferred is simply power multiplied by time: E = P × t.

这些公式有助于计算功率耗散、选择保险丝和设计电路。传输的能量即为功率乘以时间：E = P × t。

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6. Resistors in Series and Parallel | 电阻的串联与并联

When resistors are connected in series, the total resistance is the sum of the individual resistances. The same current flows through each resistor.

电阻串联时，总电阻等于各个电阻之和，且流经每个电阻的电流相同。

R_total = R₁ + R₂ + R₃ + …

For a parallel arrangement, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. The potential difference across each branch is the same.

并联时，总电阻的倒数等于各电阻倒数之和，各支路两端电势差相同。

1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …

In engineering, combining series and parallel resistors is essential for achieving desired current and voltage levels in control circuits and sensor networks.

在工程中，组合使用串联与并联电阻是实现控制电路和传感器网络中所需电流、电压水平的重要手段。

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7. Pressure in Fluids | 流体压力

Pressure is defined as force acting per unit area. In fluids, pressure increases with depth due to the weight of the fluid above.

压力定义为单位面积上所受到的力。在流体中，由于上方流体的重量，压力随深度增加而增大。

p = F / A

p is pressure in pascals (Pa), F is force (N) and A is area (m²). For a liquid column, the pressure at a depth is given by:

p 为压力，单位帕斯卡 (Pa)；F 为力 (N)；A 为面积 (m²)。对于液柱，深度处的压力由下式给出：

p = ρ × g × h

ρ (rho) is the density of the fluid in kg/m³, g is the acceleration due to gravity and h is the depth in metres. This formula explains why dam walls are thicker at the base and is used in hydraulic systems.

ρ（柔）是流体密度，单位 kg/m³；g 是重力加速度；h 为深度，单位米。该公式解释了为何水坝底部更厚，且应用于液压系统。

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8. Thermal Expansion | 热膨胀

Most materials expand when heated. The linear expansion of a solid is proportional to its original length and the temperature rise.

大多数材料受热膨胀。固体的线性膨胀量与其原始长度和温升成正比。

ΔL = α × L₀ × ΔT

ΔL is the change in length (m), α is the coefficient of linear expansion (per °C or per K), L₀ is the original length and ΔT is the temperature change. Engineers must allow for expansion in bridges, railways and pipelines to prevent buckling.

ΔL 为长度变化量 (m)；α 为线膨胀系数（每 °C 或每 K）；L₀ 为原始长度；ΔT 为温度变化量。工程师在桥梁、铁路和管道中必须预留膨胀空间以防止屈曲。

For area and volume expansion, the coefficients are approximately 2α and 3α respectively for isotropic materials.

对于面积和体积膨胀，各向同性材料的系数分别约为 2α 和 3α。

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

Simple machines such as levers, pulleys and gears provide a mechanical advantage (MA), reducing the effort needed to overcome a load.

杠杆、滑轮和齿轮等简单机械提供机械利益 (MA)，减少克服负载所需的动力。

MA = Load / Effort

It is a ratio of forces (no units). Velocity ratio (VR) compares the distance moved by the effort to the distance moved by the load.

这是一个力之比（无量纲）。速度比 (VR) 则比较动力移动的距离与负载移动的距离。

VR = distance moved by effort / distance moved by load

Efficiency links these two concepts, indicating how much of the input work is usefully transferred.

效率将这两个概念联系起来，表明输入功中有多少被有效传递。

Efficiency = (MA / VR) × 100%

or equivalently, Efficiency = (useful work output / total work input) × 100%. In real machines, efficiency is always less than 100% due to friction.

或等效为 效率 = (有用输出功 / 总输入功) × 100%。在实际机械中，由于摩擦的存在，效率始终小于 100%。

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10. Motion in Engineering (Velocity, Acceleration) | 工程运动学（速度、加速度）

Understanding motion is key to analysing mechanisms. For constant acceleration along a straight line, the following SUVAT equations apply.

理解运动是分析机构的关键。对于匀加速直线运动，可以应用以下 SUVAT 方程。

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

u is initial velocity (m/s), v is final velocity, a is acceleration (m/s²), t is time (s) and s is displacement (m). These relationships are used in designing cam profiles, robotic arms and vehicle systems.

u 为初速度 (m/s)；v 为末速度；a 为加速度 (m/s²)；t 为时间 (s)；s 为位移 (m)。这些关系用于设计凸轮轮廓、机械臂和车辆系统。

Average speed and velocity can be calculated simply as distance divided by time or displacement divided by time respectively. In circular motion the centripetal force required to keep an object moving in a circle is given by F = m v² / r, where r is the radius.

平均速率和速度可分别用路程除以时间或位移除以时间计算。在圆周运动中，维持物体做圆周运动的向心力由 F = m v² / r 给出，其中 r 为半径。

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