Year 12 CIE Engineering: Formula & Theorem Quick Reference Handbook | CIE 工程第一年：公式定理速查手册

📚 Year 12 CIE Engineering: Formula & Theorem Quick Reference Handbook | CIE 工程第一年：公式定理速查手册

This handbook provides a structured and concise collection of essential formulas, definitions and theorems required for the Year 12 CIE Engineering syllabus. Designed as a revision aid and problem‑solving companion, it covers mechanics, materials, thermal physics, electrical principles and core engineering mathematics. Use it to refresh your memory before tests or to confirm relationships while tackling practical and theoretical questions.

本手册系统汇编了 CIE 工程第一年课程的核心公式、定义和定理，覆盖力学、材料、热学、电学原理以及基础工程数学。旨在为复习提供速查工具，也为解题提供快速参考。

1. Fundamental Units and Prefixes | 基本单位与词头

All engineering quantities are expressed in SI base units or derived units. Familiarity with standard prefixes helps in managing very large or very small numerical values and avoiding conversion errors.

工程中所有物理量均使用国际单位制 (SI) 基本单位或导出单位表示。熟悉标准词头有助于处理极大或极小的数值，并能减少单位换算错误。

The seven SI base units are metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for thermodynamic temperature, mole (mol) for amount of substance and candela (cd) for luminous intensity. Most engineering work uses the first six.

七个 SI 基本单位分别是：长度—米 (m)，质量—千克 (kg)，时间—秒 (s)，电流—安培 (A)，热力学温度—开尔文 (K)，物质的量—摩尔 (mol) 以及发光强度—坎德拉 (cd)。工程中主要使用前六个。

Prefix	Symbol	Factor	词头	符号	因子
giga	G	10⁹	吉	G	10⁹
mega	M	10⁶	兆	M	10⁶
kilo	k	10³	千	k	10³
centi	c	10⁻²	厘	c	10⁻²
milli	m	10⁻³	毫	m	10⁻³
micro	μ	10⁻⁶	微	μ	10⁻⁶
nano	n	10⁻⁹	纳	n	10⁻⁹

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2. Scalar and Vector Quantities | 标量与矢量

A scalar quantity has magnitude only; a vector quantity has both magnitude and direction. In statics and dynamics, vectors are used to represent forces, velocities, accelerations and displacements.

标量仅有大小；矢量则既有大小又有方向。在静力学和动力学中，力、速度、加速度和位移均用矢量表示。

Two vectors are added using the triangle or parallelogram law. The resultant R of two vectors A and B at an angle θ is given by the cosine rule.

两矢量可按三角形法则或平行四边形法则相加。夹角为 θ 的两个矢量 A 和 B 的合矢量 R 由余弦定理给出。

R = √(A² + B² + 2AB cos θ)

In many engineering problems, a vector is resolved into two perpendicular components. For a force F at angle θ to the horizontal, the horizontal component is F cos θ and the vertical component is F sin θ.

在工程问题中，矢量常被分解为两个相互垂直的分量。与水平方向夹角为 θ 的力 F，其水平分量为 F cos θ，竖直分量为 F sin θ。

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3. Equilibrium of Forces | 力的平衡

When a body is in static equilibrium, the resultant force in any direction is zero and the sum of clockwise moments about any point equals the sum of anticlockwise moments.

当物体处于静力平衡时，任意方向上合力为零，且对任意点的顺时针力矩之和等于逆时针力矩之和。

For concurrent forces acting at a point, equilibrium requires that the vector sum of all forces is zero. This is often checked by resolving into two perpendicular directions.

对于作用在同一点的共点力，平衡条件要求所有力的矢量和为零。通常通过沿两个垂直方向分解来检验。

∑ Fₓ = 0    and    ∑ F_y = 0

If three non‑parallel forces maintain a body in equilibrium, their lines of action must be concurrent. This principle is frequently applied in structural analysis.

若三个不平行力维持物体平衡，则它们的作用线必汇交于一点。该原理在结构分析中经常使用。

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4. Moments and Couples | 力矩与力偶

The moment of a force about a point is a measure of its turning effect. It is calculated as the product of the force and the perpendicular distance from the point to the line of action of the force.

力对一点的力矩是力产生转动效应的量度。其大小为力与从该点到力作用线的垂直距离的乘积。

Moment = F × d

Where F is the force (N) and d is the perpendicular distance (m). The unit of moment is the newton metre (N m).

式中 F 为力 (N)，d 为垂直距离 (m)。力矩的单位为牛顿米 (N m)。

A couple consists of two equal, parallel and opposite forces separated by a perpendicular distance. The moment of a couple (torque) is independent of the point about which it is measured.

力偶由两个大小相等、方向相反且作用线平行的力组成，它们之间有垂直距离。力偶的力矩（转矩）与矩心选择无关。

Couple moment = F × d (where d is the perpendicular separation)

The principle of moments states that for equilibrium the sum of clockwise moments equals the sum of anticlockwise moments about any point.

力矩原理指出：平衡时，对任意点顺时针力矩之和等于逆时针力矩之和。

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5. Kinematics (Linear Motion) | 直线运动学

For motion in a straight line with constant acceleration, the four equations of motion (SUVAT) link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t).

对于匀加速直线运动，四个运动学方程 (SUVAT) 将位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t) 联系起来。

First equation: final velocity without displacement.

方程一：不含位移的末速度。

v = u + at

Second equation: displacement without final velocity.

方程二：不含末速度的位移。

s = ut + ½at²

Third equation: final velocity squared without time.

方程三：不含时间的速度平方关系。

v² = u² + 2as

Fourth equation: displacement using average velocity.

方程四：利用平均速度求位移。

s = ½(u + v)t

Acceleration is the rate of change of velocity, and the gradient of a displacement‑time graph gives velocity; the gradient of a velocity‑time graph gives acceleration. The area under a velocity‑time graph gives displacement.

加速度是速度的变化率。位移‑时间图像的斜率表示速度，速度‑时间图像的斜率表示加速度，速度‑时间图像下的面积表示位移。

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6. Newton’s Laws of Motion | 牛顿运动定律

Newton’s first law: A body remains at rest or in uniform motion in a straight line unless acted upon by a resultant external force.

牛顿第一定律：物体在不受净外力作用时，将保持静止或匀速直线运动状态。

Newton’s second law: The resultant force acting on a body is equal to the rate of change of its momentum. For constant mass, this simplifies to F = ma.

牛顿第二定律：物体所受合外力等于其动量变化率。若质量恒定，则简化为 F = ma。

F = ma

Newton’s third law: When one body exerts a force on another, the second body exerts an equal and opposite force on the first.

牛顿第三定律：当一个物体对另一物体施加力时，另一物体同时施加一个大小相等、方向相反的力。

Momentum is the product of mass and velocity and is conserved in interactions where no external resultant force acts.

动量是质量与速度的乘积。在不受净外力的相互作用中，总动量守恒。

p = mv

The impulse of a force equals the change in momentum: Ft = Δ(mv).

力的冲量等于动量的变化量：Ft = Δ(mv)。

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7. Work, Energy and Power | 功、能与功率

Work is done when a force moves its point of application in the direction of the force. The work done W is the product of the force and the distance moved in the direction of the force.

当力的作用点沿力的方向移动时，力就做了功。功 W 等于力与沿力方向移动距离的乘积。

W = Fd cos θ

Gravitational potential energy (GPE) gained by a mass lifted through a vertical height h is mgh. Kinetic energy (KE) of a moving body is ½mv².

质量为 m 的物体被竖直提升高度 h 所增加的重力势能 (GPE) 为 mgh。运动物体的动能 (KE) 为 ½mv²。

GPE = mgh    and    KE = ½mv²

The principle of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In the absence of friction, the total mechanical energy remains constant.

能量守恒原理指出，能量既不能创生也不能消灭，只能从一种形式转化为另一种。在没有摩擦的情况下，总机械能保持恒定。

Power is the rate of doing work or of transferring energy. It can be expressed as force × velocity for a body moving at constant speed.

功率是做功或传递能量的速率。对于匀速运动的物体，功率可以表示为力与速度的乘积。

P = W/t = Fv

Efficiency is the ratio of useful output power (or energy) to input power (or energy), often expressed as a percentage.

效率是有用输出功率（或能量）与输入功率（或能量）之比，常以百分比表示。

Efficiency = (Useful output / Total input) × 100%

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8. Stress and Strain | 应力与应变

Tensile stress σ is defined as the applied force divided by the original cross‑sectional area of the material. Compressive stress is calculated using the same relationship.

拉伸应力 σ 定义为施加的力除以材料的原始截面积。压缩应力使用同样的关系式计算。

σ = F / A

Tensile strain ε is the extension per unit original length. It is a dimensionless ratio.

拉伸应变 ε 是单位原始长度的伸长量，为一个无量纲比值。

ε = ΔL / L₀

Where ΔL is the change in length and L₀ is the original length. The unit of stress is the pascal (Pa), where 1 Pa = 1 N m⁻².

式中 ΔL 为长度变化量，L₀ 为原始长度。应力的单位是帕斯卡 (Pa)，1 Pa = 1 N m⁻²。

A stress‑strain graph for a ductile material shows an initial linear region, followed by yielding, plastic deformation and ultimately fracture. The elastic limit marks the maximum stress for which the material returns to its original length upon unloading.

延性材料的应力‑应变曲线显示初始线性阶段，随后出现屈服、塑性变形直至断裂。弹性极限是卸载后材料能够恢复原始长度的最大应力。

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9. Young’s Modulus | 杨氏模量

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

杨氏模量 E 是衡量固体材料刚度的物理量。在材料的比例（线性）极限内，它等于拉伸应力与拉伸应变之比。

E = σ / ε

Substituting the definitions of stress and strain gives a practical form of the relationship that involves the applied force, original length, cross‑sectional area and extension.

代入应力和应变的定义，可以得到一个包含施加力、原始长度、截面积和伸长量的实用关系式。

E = (F L₀) / (A ΔL)

Young’s modulus has the same units as stress, Pa or N m⁻², and is constant for a given material under elastic conditions. A steep initial gradient on a stress‑strain graph corresponds to a high Young’s modulus and a stiff material.

杨氏模量的单位与应力相同，为 Pa 或 N m⁻²。对于给定材料，在弹性范围内其值为常数。应力‑应变曲线初始斜率较陡表示杨氏模量高，材料刚度大。

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10. Thermal Expansion and Heat Transfer | 热膨胀与传热

Linear thermal expansion describes the change in length of a solid as its temperature changes. The increase in length ΔL is proportional to the original length L₀ and the temperature change ΔT.

线热膨胀描述固体长度随温度变化的关系。长度增量 ΔL 与原始长度 L₀ 和温度变化 ΔT 成正比。

ΔL = α L₀ ΔT

Where α is the coefficient of linear expansion of the material (unit K⁻¹ or °C⁻¹). The final length is L₀ + ΔL.

式中 α 为材料的线膨胀系数（单位为 K⁻¹ 或 °C⁻¹）。最终长度为 L₀ + ΔL。

Heat transfer occurs by conduction, convection and radiation. Conduction through a solid of cross‑sectional area A, thickness d and thermal conductivity k is described by a steady‑state rate equation.

传热通过传导、对流和辐射三种方式进行。对于截面积为 A、厚度为 d、热导率为 k 的固体，稳态导热可用以下速率方程描述。

Q/t = k A (ΔT / d)

Where Q/t is the rate of heat transfer and ΔT is the temperature difference across the material. In many engineering contexts, expansion allowances such as expansion joints are essential to prevent structural damage.

式中 Q/t 为传热速率，ΔT 为材料两端的温差。在许多工程场合，膨胀补偿（如膨胀缝）是防止结构损坏的关键措施。

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11. Ohm’s Law and Circuits | 欧姆定律与电路

Ohm’s law relates the potential difference V across a conductor to the current I flowing through it and its resistance R. For ohmic materials, the relationship is linear at constant temperature.

欧姆定律将导体两端的电势差 V、流过导体的电流 I 及其电阻 R 联系在一起。对欧姆材料，在恒温下关系呈线性。

V = IR

Electrical power P dissipated in a resistor can be expressed in three equivalent forms, depending on which quantities are known.

电阻消耗的电功率 P 可根据已知量用三种等价形式表示。

P = VI = I²R = V²/R

Resistors in series: the total resistance is the sum of individual resistances. Resistors in parallel: the reciprocal of the total resistance equals the sum of the reciprocals of the individual resistances.

串联电阻：总电阻等于各电阻之和。并联电阻：总电阻的倒数等于各电阻倒数之和。

Series: R_total = R₁ + R₂ + …

Parallel: 1/R_total = 1/R₁ + 1/R₂ + …

Kirchhoff’s current law (KCL): the total current entering a junction equals the total current leaving it. Kirchhoff’s voltage law (KVL): the sum of the emfs in any closed loop equals the sum of the potential drops across the resistors in that loop.

基尔霍夫电流定律 (KCL)：流入节点的总电流等于流出节点的总电流。基尔霍夫电压定律 (KVL)：任一闭合回路中，电动势之和等于各电阻上电势降落之和。

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12. Engineering Mathematics (Key Formulas) | 工程数学（关键公式）

Engineers routinely apply geometry, trigonometry and mensuration. The following formulas are indispensable for calculating areas, volumes and components of forces.

工程师经常用到几何、三角和测量公式。以下公式对于计算面积、体积和力的分力至关重要。

Pythagoras’ theorem relates the sides of a right‑angled triangle.

勾股定理联系直角三角形的三条边。

a² + b² = c²

Basic trigonometric ratios (SOH‑CAH‑TOA) for a right‑angled triangle with angle θ:

直角三角形中角 θ 的基本三角比 (SOH‑CAH‑TOA)：

sin θ = opposite/hypotenuse    cos θ = adjacent/hypotenuse    tan θ = opposite/adjacent

The sine rule and cosine rule extend trigonometry to non‑right‑angled triangles.

正弦定理和余弦定理将三角学推广到非直角三角形。

a/sin A = b/sin B = c/sin C

a² = b² + c² − 2bc cos A

Area formulas: rectangle (length × width), triangle (½ × base × height), circle (πr²) and area of a sector (½ r² θ with θ in radians).

面积公式：矩形（长 × 宽）、三角形（½ × 底 × 高）、圆（πr²）以及扇形面积（½ r² θ，θ 以弧度计）。

Volumes: cuboid (length × width × height), prism (area of cross‑section × length), cylinder (πr²h), sphere (⁴⁄₃ πr³) and cone (⅓ πr²h). Surface area of a sphere is 4πr².

体积：长方体（长 × 宽 × 高）、棱柱（截面积 × 长）、圆柱 (πr²h)、球体 (⁴⁄₃ πr³) 和圆锥 (⅓ πr²h)。球体的表面积为 4πr²。

Conversion between degrees

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