Year 11 CIE Engineering: Formula & Theorem Quick Reference Handbook | CIE工程公式定理速查手册

📚 Year 11 CIE Engineering: Formula & Theorem Quick Reference Handbook | CIE工程公式定理速查手册

This rapid-reference handbook brings together the essential formulas and theorems that every Year 11 CIE Engineering student needs at their fingertips. From moments and motion to materials and electricity, each section provides a concise definition, key equations, and typical applications to support effective revision and problem-solving.

本速查手册汇集了每个Year 11 CIE工程学生需要掌握的必备公式和定理。从力矩、运动到材料与电学，每一节都提供简明定义、关键方程和典型应用，支持高效复习和解题。

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1. 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 magnitude of the force and the perpendicular distance from the pivot to the line of action of the force. The principle of moments states that for a system in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.

力矩是力围绕一个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。力矩原理指出，对于处于转动平衡的系统，顺时针力矩之和等于逆时针力矩之和。

• Moment formula:

M = F × d

where M is the moment (N m), F is the force (N), and d is the perpendicular distance from the pivot (m).

其中 M 为力矩（N m），F 为力（N），d 为距支点的垂直距离（m）。

• Equilibrium conditions:

For a body to be in static equilibrium, two conditions must be satisfied: the resultant force in any direction must be zero, and the resultant moment about any point must be zero.

物体要处于静力平衡，必须满足两个条件：任意方向上的合力为零，且对任意点的合力矩为零。

ΣF = 0  and  ΣM = 0

In beam and lever problems, taking moments about a support often simplifies calculations.

在梁和杠杆问题中，对某一支撑点取矩通常可以简化计算。

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2. Forces and Motion | 力与运动

Newton's second law relates the net force acting on an object to its mass and acceleration. The equations of uniformly accelerated motion (SUVAT) describe linear motion when acceleration is constant.

牛顿第二定律将作用在物体上的合力与其质量和加速度联系起来。匀加速直线运动方程（SUVAT）描述加速度恒定时物体的直线运动。

• Newton's Second Law:

F = m × a

F is resultant force (N), m is mass (kg), a is acceleration (m s⁻²).

F 为合力（N），m 为质量（kg），a 为加速度（m s⁻²）。

• SUVAT equations (valid when a = constant):

v = u + at

s = ut + ½at²

v² = u² + 2as

where u is initial velocity, v is final velocity (m s⁻¹), a is acceleration (m s⁻²), t is time (s), s is displacement (m).

其中 u 为初速度，v 为末速度（m s⁻¹），a 为加速度（m s⁻²），t 为时间（s），s 为位移（m）。

• Weight:

W = m × g

g is gravitational field strength (≈ 9.8 N kg⁻¹ on Earth).

g 为重力场强度（地球表面约为 9.8 N kg⁻¹）。

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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 an object in the direction of the force. Power is the rate of doing work or transferring energy.

能量是做功的本领。当力使物体沿力的方向移动时，力就做了功。功率是做功或能量转移的速率。

• Work done:

W = F × d

W is work (J), F is force (N), d is distance moved in direction of force (m).

W 为功（J），F 为力（N），d 为沿力方向移动的距离（m）。

• Kinetic energy (KE):

Eₖ = ½mv²

• Gravitational potential energy (GPE):

Eₚ = mgh

h is vertical height (m).

h 为垂直高度（m）。

• Power:

P = W / t   and   P = F × v

P is power (W), v is velocity (m s⁻¹) for a constant force applied in direction of motion.

P 为功率（W），在力与运动方向一致且力恒定时，可用 P = Fv。

• Efficiency:

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

η = (useful power output / total power input) × 100%

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4. Stress, Strain and Elasticity | 应力、应变与弹性

When a material is subjected to a load, it experiences stress and deforms, producing strain. Hooke's law describes the linear relationship between force and extension for many materials within their elastic limit.

当材料承受载荷时，它会产生应力并发生变形，产生应变。胡克定律描述了许多材料在弹性极限内力与伸长量之间的线性关系。

• Stress (σ):

σ = F / A

σ is stress (Pa or N m⁻²), F is applied force (N), A is cross-sectional area (m²).

σ 为应力（Pa 或 N m⁻²），F 为施加的力（N），A 为横截面积（m²）。

• Strain (ε):

ε = ΔL / L₀

ε is strain (no units), ΔL is change in length (m), L₀ is original length (m).

ε 为应变（无单位），ΔL 为长度变化量（m），L₀ 为原始长度（m）。

• Young's modulus (E):

E = σ / ε

E is a measure of stiffness, valid within the limit of proportionality.

E 是刚度的衡量指标，仅在比例极限内有效。

• Hooke's Law:

F = k × x

k is the spring constant (N m⁻¹), x is extension (m).

k 为弹簧常数（N m⁻¹），x 为伸长量（m）。

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5. Mechanical Advantage and Velocity Ratio | 机械效益与速度比

Simple machines make work easier by multiplying force or changing its direction. Mechanical advantage (MA) indicates how much a machine multiplies the effort force, while velocity ratio (VR) compares the distances moved by effort and load.

简单机械通过放大作用力或改变其方向使工作更省力。机械效益（MA）表示机械放大作用力的倍数，而速度比（VR）则比较作用力移动的距离与负载移动的距离。

• Mechanical Advantage:

MA = Load / Effort

Both measured in newtons; MA has no unit. In an ideal machine with no friction, MA = VR.

负载和作用力均以牛顿为单位；MA 无单位。在无摩擦的理想机械中，MA = VR。

• Velocity Ratio:

VR = distance moved by effort / distance moved by load in the same time

/ load arm length. For a pulley system, VR = number of rope segments supporting the load.

对于杠杆，VR = 力臂长度 / 重臂长度。对于滑轮组，VR = 支持负载的绳索段数。

• Efficiency from MA and VR:

η = (MA / VR) × 100%

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6. Gears and Pulleys | 齿轮与滑轮

Gears transmit rotary motion and torque between shafts. Pulleys change the direction of a force and provide mechanical advantage in lifting systems.

齿轮在轴之间传递旋转运动和扭矩。滑轮改变力的方向，并在提升系统中提供机械效益。

• Gear ratio:

Gear ratio = number of teeth on driven gear / number of teeth on driver gear

A ratio greater than 1 gives speed reduction and torque multiplication. A ratio less than 1 increases speed.

比值大于 1 时减速增扭；小于 1 时增速。

• Speed ratio of meshing gears:

N₁ / N₂ = T₂ / T₁

N₁, N₂ are speeds (rpm) of driver and driven; T₁, T₂ are tooth counts.

N₁, N₂ 分别为主动轮和从动轮的转速（rpm）；T₁, T₂ 为齿数。

• Pulley systems (block and tackle):

VR = number of rope strands supporting the moving block

Tension in each strand is approximately equal; total effort is shared among the strands.

各绳索张力近似相等；总作用力由多股绳索分担。

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7. Fluid Systems | 流体系统

Fluid power systems use liquids (hydraulics) or gases (pneumatics) to transmit force. Key principles include pressure in a static fluid and flow continuity.

流体动力系统利用液体（液压）或气体（气动）来传递力。关键原理包括静止流体中的压力和流动连续性。

• Pressure:

P = F / A

P is pressure (Pa), F is force normal to surface (N), A is area (m²).

P 为压强（Pa），F 为垂直于表面的力（N），A 为面积（m²）。

• Hydrostatic pressure:

P = h ρ g

h is depth of fluid (m), ρ is fluid density (kg m⁻³), g is gravitational field strength (N kg⁻¹).

h 为液体深度（m），ρ 为液体密度（kg m⁻³），g 为重力场强度（N kg⁻¹）。

• Pascal's principle in hydraulic systems:

F₁ / A₁ = F₂ / A₂

A small force applied to a small piston can create a larger force on a larger piston, trading distance for force.

施加在小活塞上的小力可以转化为大活塞上的大力，以行程换取力的放大。

• Flow rate (continuity):

Q = A × v

te (m³ s⁻¹), v is average flow velocity (m s⁻¹). For an incompressible fluid, A₁v₁ = A₂v₂.

Q 为体积流量（m³ s⁻¹），v 为平均流速（m s⁻¹）。对于不可压缩流体，A₁v₁ = A₂v₂。

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8. Electrical Principles | 电学原理

Engineering often involves electrical circuits for sensing, control and actuation. Ohm's law and Kirchhoff's laws are fundamental for analysing DC circuits.

工程中常涉及用于传感、控制和驱动的电路。欧姆定律和基尔霍夫定律是分析直流电路的基础。

• Ohm's Law:

V = I × R

V is voltage (V), I is current (A), R is resistance (Ω).

V 为电压（V），I 为电流（A），R 为电阻（Ω）。

• Resistors in series:

Rₜ = R₁ + R₂ + R₃ + ...

• Resistors in parallel:

1/Rₜ = 1/R₁ + 1/R₂ + 1/R₃ + ...

• Electrical power:

P = I × V = I²R = V²/R

P is power (W).

P 为功率（W）。

• Kirchhoff's current law (KCL):

At any junction, the sum of currents entering equals the sum of currents leaving.

在任一节点，流入电流之和等于流出电流之和。

• Kirchhoff's voltage law (KVL):

In a closed loop, the sum of emf equals the sum of voltage drops.

在闭合回路中，电动势之和等于电压降之和。

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9. Thermal Effects | 热效应

When materials are heated, they expand and absorb energy. These thermal properties are crucial in designing structures and machines that operate across temperature changes.

当材料受热时，会发生膨胀并吸收能量。这些热学性质在设计承受温度变化的结构和机器时至关重要。

• Linear thermal expansion:

ΔL = α L₀ Δθ

ΔL is change in length (m), α is coefficient of linear expansion (K⁻¹), L₀ is original length (m), Δθ is temperature change (K or °C).

ΔL 为长度变化量（m），α 为线胀系数（K⁻¹），L₀ 为原始长度（m），Δθ 为温度变化量（K 或 °C）。

• Heat energy (no phase change):

Q = m c Δθ

Q is heat energy (J), m is mass (kg), c is specific heat capacity (J kg⁻¹ K⁻¹).

Q 为热量（J），m 为质量（kg），c 为比热容（J kg⁻¹ K⁻¹）。

• Ideal gas law (for pneumatics):

pV = nRT

olume (m³), n is amount of gas (mol), R = 8.31 J K⁻¹ mol⁻¹, T is absolute temperature (K).

p 为压强（Pa），V 为体积（m³），n 为气体物质的量（mol），R = 8.31 J K⁻¹ mol⁻¹，T 为热力学温度（K）。

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10. Forces in Structures | 结构中的力

Structures such as frameworks and trusses carry loads through tension and compression in members. Analysing these internal forces relies on vector addition and static equilibrium.

框架和桁架等结构通过构件的拉伸和压缩来承担载荷。分析这些内力依赖于矢量加法和静力平衡。

• Resolution of a force:

A force F at angle θ to a reference axis has components:

力 F 与参考轴成 θ 角时的分量为：

Fₓ = F cosθ,   Fᵧ = F sinθ

• Resultant of two perpendicular forces:

R = √Fₓ² + Fᵧ²,   direction θ = tan⁻¹(Fᵧ / Fₓ)

• Triangle of forces:

If three forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the sides of a triangle taken in order.

若作用于一点的三个力平衡，则它们可以按顺序用三角形的三条边来表示大小和方向。

• Truss analysis – method of joints (simplified):

0. Assume unknown member forces to be in tension; a negative result indicates compression.

在每个节点，ΣFₓ = 0 且 ΣFᵧ = 0。假设未知杆件内力为拉力；若结果为负，则表示受压。

• Factor of Safety (FoS):

FoS = ultimate stress / allowable working stress

This ensures structures can support loads well beyond those expected in normal use.

这确保结构能够支撑远超正常使用预期的载荷。

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Published by TutorHao | Engineering Revision Series | aleveler.com

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