Cambridge IGCSE Engineering Formula Quick Reference Handbook | 剑桥IGCSE工程公式定理速查手册

📚 Cambridge IGCSE Engineering Formula Quick Reference Handbook | 剑桥IGCSE工程公式定理速查手册

This quick-reference handbook gathers all the essential formulas, principles, and theorems you need for the Year 11 Cambridge IGCSE Engineering syllabus. Use these pages to reinforce your problem-solving skills and build confidence before the examination.

本速查手册汇集了 Year 11 剑桥 IGCSE 工程课程所需的所有核心公式、原理与定理。通过这些内容巩固解题能力，考前倍增信心。

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1. Forces and Equilibrium | 力与平衡

A force is a push or pull that can change the motion or shape of an object. When all forces acting on a body cancel out, the body is in equilibrium, meaning it remains stationary or moves with constant velocity.

力是能改变物体运动状态或形状的推或拉。当作用在一个物体上的所有力相互抵消时，物体处于平衡状态，即保持静止或匀速直线运动。

• Resultant force F = m × a (Newton’s second law) — the net force equals mass times acceleration.
• 合力 F = m × a（牛顿第二定律）—— 净力等于质量乘以加速度。
• For equilibrium: ΣF_horizontal = 0 and ΣF_vertical = 0. The total upward forces equal the total downward forces; total leftward forces equal total rightward forces.
• 平衡条件：ΣF_水平 = 0 且 ΣF_垂直 = 0。向上的合力等于向下的合力；向左的合力等于向右的合力。

Weight W = m × g

Weight is the gravitational force on a mass. g = 9.8 m/s² on Earth (use 10 m/s² for estimates).

重量是作用在质量上的重力。地球上 g = 9.8 m/s²（估算时可取 10 m/s²）。

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2. Motion | 运动

Kinematics describes how objects move without considering the forces that cause the motion. The following equations apply for uniform acceleration in a straight line.

运动学描述物体如何运动，而不考虑引起运动的力。以下方程适用于匀加速直线运动。

Key variables: u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement.

关键变量：u = 初速度，v = 末速度，a = 加速度，t = 时间，s = 位移。

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

s = (u + v) / 2 × t

Use these equations only when acceleration is constant. Select the equation that includes the three known quantities plus the one unknown you need.

仅当加速度恒定时使用这些方程。选择包含三个已知量和一个未知量的方程进行求解。

Average speed = total distance / total time. For non-uniform motion, use a velocity–time graph: the gradient gives acceleration, the area under the graph gives displacement.

平均速率 = 总路程 / 总时间。对于非匀速运动，使用速度–时间图：斜率表示加速度，图下面积代表位移。

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3. Energy, Work and Power | 能量、功和功率

Energy is the capacity to do work. Work is done when a force moves its point of application in the direction of the force.

能量是做功的本领。当力使其作用点沿力的方向移动时，力就做了功。

Work done W = F × d

where F is the force in newtons and d is the distance moved in the direction of the force (m). Work is measured in joules (J).

式中 F 为力（牛顿），d 为沿力方向移动的距离（米）。功的单位是焦耳（J）。

Gravitational potential energy GPE = m × g × h

Here h is the vertical height gained. Kinetic energy:

其中 h 是获得的垂直高度。动能：

Kinetic energy KE = ½ m v²

Conservation of energy: In a closed system, total energy is constant. Energy can be transformed from one form to another, but cannot be created or destroyed.

能量守恒：在封闭系统中，总能量保持不变。能量可以从一种形式转换为另一种形式，但不会被创造或消灭。

Power P = W / t

Power is the rate of doing work, measured in watts (W), where 1 W = 1 J/s. Also, for a constant force moving at constant speed v: P = F × v.

功率是做功的速率，单位为瓦特（W），1 W = 1 J/s。对于以恒定速度 v 移动的恒力：P = F × v。

Efficiency = (useful energy output / total energy input) × 100%

Efficiency can also be expressed as (useful power output / total power input). A machine can never have 100% efficiency because of friction and heat losses.

效率也可表示为（有用输出功率 / 总输入功率）。由于摩擦和热损耗，任何机器的效率都不可能达到 100%。

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4. Moments and Levers | 力矩与杠杆

A moment is the turning effect of a force about a pivot. It depends on 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。它取决于力的大小以及从支点到力作用线的垂直距离。

Moment = F × d_{perpendicular}

The unit of a moment is newton metre (N m). For an object in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot.

力矩的单位是牛顿·米（N m）。物体处于转动平衡时，绕任何支点的顺时针力矩之和等于逆时针力矩之和。

Levers: First-order lever has the pivot between load and effort (e.g. see-saw). Second-order lever has the load between pivot and effort (e.g. wheelbarrow). Third-order lever has the effort between pivot and load (e.g. tweezers).

杠杆：一级杠杆的支点在负载与施力之间（如跷跷板）；二级杠杆的负载在支点与施力之间（如手推车）；三级杠杆的施力在支点与负载之间（如镊子）。

The principle of moments works for all levers: Load × load arm = Effort × effort arm. This relationship underpins mechanical advantage.

力矩原理适用于所有杠杆：负载 × 负载臂 = 施力 × 施力臂。这一关系是机械利益的基础。

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5. Stress and Strain | 应力与应变

When a force is applied to a material, it experiences stress and strain. Knowing these helps engineers choose the right material.

当力作用于材料时，材料会承受应力和应变。了解这些有助于工程师选择正确的材料。

Stress σ = F / A

Stress is the force per unit cross-sectional area. It is measured in pascals (Pa) or N/m². 1 Pa = 1 N/m².

应力是单位横截面积上的力，单位为帕斯卡（Pa）或 N/m²。1 Pa = 1 N/m²。

Strain ε = ΔL / L₀

Strain is the extension per unit original length. It is a ratio and has no units. ΔL is the change in length, L₀ is the original length.

应变是单位原长的伸长量，是一个比值，没有单位。ΔL 为长度变化量，L₀ 为原始长度。

For many materials, stress is proportional to strain up to the limit of proportionality. This relationship is Hooke’s Law:

对于许多材料，在比例极限以内，应力与应变成正比。这一关系即胡克定律：

F = k × x

where F is the applied force, x is the extension (or compression), and k is the spring constant (stiffness). For a wire, the spring constant depends on the material, length, and cross-sectional area.

其中 F 是施加的力，x 是伸长量（或压缩量），k 是弹簧常数（刚度）。对于金属丝，弹簧常数取决于材料、长度和横截面积。

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6. Young Modulus and Material Properties | 杨氏模量与材料性质

Young modulus E measures the stiffness of a material and is defined as the ratio of stress to strain within the elastic limit.

杨氏模量 E 衡量材料的刚度，定义为弹性极限内应力与应变之比。

E = σ / ε = (F / A) / (ΔL / L₀)

Its unit is pascals (Pa). A high E means the material is stiff (e.g. steel), a low E means it is flexible (e.g. rubber).

单位为帕斯卡（Pa）。E 值高表示材料刚性强（如钢），E 值低表示材料柔韧（如橡胶）。

Other key mechanical properties: elastic limit – the maximum stress that does not cause permanent deformation; yield point – stress at which material begins to deform plastically; ultimate tensile strength (UTS) – maximum stress before fracture; ductility – ability to be drawn into wire; malleability – ability to be hammered into thin sheets; hardness – resistance to indentation or scratching.

其他关键力学性能：弹性极限——不产生永久变形的最大应力；屈服点——材料开始发生塑性变形的应力；抗拉强度 (UTS)——断裂前的最大应力；延展性——能被拉成细丝的能力；可锻性——能被锤成薄片的能力；硬度——抵抗压入或刮擦的能力。

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7. Electricity Basics | 电学基础

Electric current is the rate of flow of electric charge. Charge is measured in coulombs (C), current in amperes (A).

电流是电荷流动的速率。电荷的单位是库仑 (C)，电流的单位是安培 (A)。

I = Q / t

Where I is current, Q is charge, t is time. Voltage (potential difference) is the energy transferred per unit charge:

其中 I 为电流，Q 为电荷，t 为时间。电压（电势差）是每单位电荷传递的能量：

V = W / Q

V is measured in volts (V), W in joules (J). Resistance R opposes current flow and is measured in ohms (Ω).

V 的单位是伏特 (V)，W 的单位是焦耳 (J)。电阻 R 阻碍电流流动，单位是欧姆 (Ω)。

For many components, the relationship between voltage, current and resistance is given by Ohm’s law.

对于许多元件，电压、电流和电阻之间的关系由欧姆定律给出。

V = I × R

Ohm’s law holds when temperature is constant. A component that obeys Ohm’s law is called an ohmic conductor.

当温度恒定时欧姆定律成立。遵守欧姆定律的元件称为欧姆导体。

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8. Electrical Power and Energy | 电功率与电能

Electrical power is the rate at which electrical energy is converted into other forms (heat, light, motion).

电功率是电能转化为其他形式（热、光、运动）的速率。

P = I × V

Power is measured in watts (W). Using Ohm’s law, alternative forms are:

功率的单位是瓦特 (W)。利用欧姆定律，可得其他形式：

P = I² × R and P = V² / R

The energy transferred by an appliance is:

用电器传递的能量为：

E = P × t = I × V × t

Energy is often expressed in joules (J), but for domestic consumption kilowatt-hours (kW·h) are used: 1 kW·h = 3.6 × 10⁶ J.

能量通常用焦耳 (J) 表示，但在家庭用电中使用千瓦时 (kW·h)：1 kW·h = 3.6 × 10⁶ J。

Cost of electricity = power (kW) × time (h) × cost per kW·h. Fuse rating should be slightly higher than the normal operating current of the appliance.

电费 = 功率 (kW) × 时间 (h) × 每千瓦·时单价。保险丝的额定电流应稍高于电器正常工作电流。

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9. Pneumatics and Hydraulics | 气动与液压

Both systems transmit force using a fluid: pneumatics uses compressed air, hydraulics uses incompressible liquid (oil).

两种系统都利用流体传递力：气动使用压缩空气，液压使用不可压缩的液体（油）。

The principle of transmission of fluid pressure (Pascal’s principle): Pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of the container.

流体压力传递原理（帕斯卡原理）：施加在密闭流体上的压力会大小不变地传递到流体的各部分及容器壁上。

p = F₁ / A₁ = F₂ / A₂

Where p is pressure (Pa), F is force (N), A is cross-sectional area (m²). In a hydraulic system, a small force applied to a small piston can produce a large force on a larger piston, because the pressure is the same.

其中 p 为压强 (Pa)，F 为力 (N)，A 为横截面积 (m²)。在液压系统中，施加在小活塞上的小力可以因压力相同而在大活塞上产生大力。

However, the distance moved by the larger piston is smaller (energy conservation: work input = work output, ignoring losses). The volume of fluid displaced is constant: A₁ d₁ = A₂ d₂.

然而，大活塞移动的距离较小（能量守恒：输入功 = 输出功，忽略损耗）。排开的液体体积恒定：A₁ d₁ = A₂ d₂。

Pneumatic systems rely on similar principles but use air, which is compressible. They are often used for quick, light-duty automation. Hydraulics are used for heavy-duty applications such as car brakes and jacks.

气动系统依赖相似原理，但使用可压缩的空气，常用于快速、轻载的自动化。液压则用于重载场合，如汽车制动器和千斤顶。

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10. Mechanical Advantage and Velocity Ratio | 机械利益与速比

Mechanical advantage (MA) tells you how much a machine multiplies the effort force. Velocity ratio (VR) compares the distance moved by the effort to the distance moved by the load in the same time.

机械利益 (MA) 表示机器将施力放大了多少倍。速比 (VR) 是比较在同一时间内施力移动的距离与负载移动的距离。

MA = Load / Effort

No units, because it is a ratio. For an ideal machine with 100% efficiency, MA = VR. In practice, MA is always less than VR because of friction.

无单位，因为它是比值。对于效率 100% 的理想机械，MA = VR。实际上，由于摩擦，MA 总小于 VR。

VR = distance moved by effort / distance moved by load

For levers, VR can be calculated from the lengths of the effort arm and load arm: VR = effort arm / load arm. For pulleys, VR = number of rope sections supporting the load. For gears, VR = number of teeth on driven gear / number of teeth on driving gear; also inversely related to rotational speeds.

对于杠杆，VR 可由施力臂与负载臂长度计算：VR = 施力臂 / 负载臂。对于滑轮，VR = 支撑负载的绳索段数。对于齿轮，VR = 从动齿轮齿数 / 主动齿轮齿数；也与转速成反比。

Efficiency links these two ratios:

效率将这两个比值联系起来：

Efficiency = (MA / VR) × 100%

This formula is essential for analysing the performance of any mechanism, from simple winches to complex gear trains.

此公式对于分析任何机构（从简单的绞盘到复杂的齿轮系）的性能至关重要。

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11. Heat and Temperature | 热与温度

Heat is a form of energy that flows from a hotter body to a colder one. Temperature measures the average kinetic energy of the particles in a substance.

热是一种从较热物体流向较冷物体的能量形式。温度衡量物质内粒子的平均动能。

The amount of heat energy required to change the temperature of a substance depends on its mass, specific heat capacity, and the temperature change:

改变物质温度所需的热量取决于其质量、比热容和温度变化：

Q = m × c × Δθ

Where Q is heat energy (J), m is mass (kg), c is specific heat capacity (J/kg·°C), Δθ is temperature change (°C). For melting or boiling at constant temperature, latent heat is involved:

其中 Q 为热量 (J)，m 为质量 (kg)，c 为比热容 (J/kg·°C)，Δθ 为温度变化 (°C)。在恒定温度下熔化或沸腾时涉及潜热：

Q = m × L

where L is specific latent heat (J/kg). Thermal expansion is important in engineering design: linear expansion ΔL = α L₀ Δθ, where α is the coefficient of linear expansion.

其中 L 为比潜热 (J/kg)。热膨胀在工程设计中十分重要：线膨胀 ΔL = α L₀ Δθ，α 为线膨胀系数。

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12. Common Engineering Applications of Formulas | 工程中公式的常见应用

These formulas are not isolated; they are combined in real-world engineering. Below is a summary table of common scenarios.

这些公式并非孤立，而是在实际工程中综合运用。下表汇总了常见应用情境。

Application	Scenario	Relevant Formulas
Bridge support	均衡负载支撑	Moments, Equilibrium
Car braking	液压制动系统	P = F/A, Work, KE → heat
Electric motor	电动机选型	P=IV, Efficiency, T = F×d
Crane lifting	起重机吊升	W = Fd, GPE, MA, Pulleys
Material selection	选材判断	σ, ε, E, UTS, Ductility

Remember that units must be consistent: always convert to SI units (metres, kilograms, seconds, amperes) before substituting into formulas.

记住单位必须一致：代入公式前务必转换为国际单位制（米、千克、秒、安培）。

Practise rearranging formulas to solve for any unknown variable. Draw clear diagrams and label forces, distances, and angles. This handbook is a revision companion — use it alongside your class notes and past papers.

练习公式变形以求解任意未知量。绘制清晰图示，标出力、距离和角度。本手册为复习伴侣——请配合课堂笔记和历年真题使用。

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