Year 11 OCR Engineering: Formula & Theorem Quick Reference Handbook | OCR 11年级工程：公式定理速查手册

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

This quick reference handbook compiles every essential formula and theorem required for the Year 11 OCR Engineering syllabus. Use it to revise key principles from mechanics, electronics and materials science. Each entry presents the equation, defines the symbols in straightforward terms, and explains when to apply it. Keep this document close while working through past-paper questions and practical design challenges.

这本速查手册汇总了 OCR 11年级工程课程中所有必考公式与核心定理。你可以用它复习力学、电子学和材料科学的关键原理。每条公式都给出了清晰的符号定义，并说明适用场合。做真题卷和设计实践题时，把这份资料放在手边，能帮你快速查漏补缺。

1. Stress & Strain | 应力与应变

Stress (σ) measures the internal force per unit area within a material. Strain (ε) describes how much a material deforms relative to its original length. These two quantities are fundamental to understanding material behaviour under load.

应力（σ）衡量材料内部单位面积所受的内力。应变（ε）描述材料相对于原始长度的变形程度。这两个量是理解材料在载荷下行为的基础。

σ = F / A

σ – stress in pascals (Pa), F – applied force in newtons (N), A – cross‑sectional area in square metres (m²).

σ – 应力，单位帕斯卡（Pa）；F – 作用力，单位牛顿（N）；A – 横截面积，单位平方米（m²）。

ε = ΔL / L₀

ε – strain (dimensionless), ΔL – change in length (m), L₀ – original length (m).

ε – 应变（无量纲）；ΔL – 长度变化量（m）；L₀ – 原始长度（m）。

In OCR problems, strain is often expressed as a percentage – simply multiply ε by 100.

在OCR考题中，应变常用百分数表示，只需将ε乘以100。

2. Hooke's Law & the Spring Constant | 胡克定律与弹性系数

Hooke's Law states that the extension of an elastic object is directly proportional to the applied force, provided the elastic limit is not exceeded. It is the foundation for analysing springs, wires and many structural members.

胡克定律指出，在不超过弹性极限的前提下，弹性物体的伸长量与所施加的力成正比。这是分析弹簧、金属丝和许多结构构件的基础。

F = k × x

F – force (N), k – spring constant or stiffness (N/m), x – extension (m).

F – 力（N）；k – 弹性系数或刚度（N/m）；x – 伸长量（m）。

The spring constant k is determined from the gradient of a force‑extension graph. For springs in parallel, k_total = k₁ + k₂; for springs in series, 1/k_total = 1/k₁ + 1/k₂.

弹性系数k可以从力-伸长图线的斜率得到。弹簧并联时 k_总 = k₁ + k₂；弹簧串联时 1/k_总 = 1/k₁ + 1/k₂。

Stored elastic potential energy in a spring obeying Hooke's Law is given by the area under the force‑extension graph.

遵守胡克定律的弹簧所储存的弹性势能等于力-伸长图线下的面积。

Eₑ = ½ × F × x = ½ × k × x²

Use the first form when force and extension are known directly; the second form connects energy to stiffness and extension squared.

已知力和伸长量时用第一种形式；第二种形式将能量与刚度及伸长量的平方关联起来。

3. Moments & the Principle of Moments | 力矩与力矩原理

A moment is the turning effect of a force about a pivot. The principle of moments states that for a system in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.

力矩是力绕支点产生的转动效应。力矩原理指出，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。

M = F × d

M – moment (Nm), F – force (N), d – perpendicular distance from the pivot to the line of action of the force (m).

M – 力矩（N·m）；F – 力（N）；d – 支点到力作用线的垂直距离（m）。

Centre of gravity is a key concept: the point through which the entire weight of an object appears to act. When drawing free‑body diagrams for beams and levers, always mark the perpendicular distance clearly.

重心是一个关键概念：整个物体的重量好像集中作用于这一点。画梁和杠杆的受力图时，一定要清晰标注出垂直距离。

To solve lever systems: identify pivot, sum clockwise moments, sum anticlockwise moments, set them equal, and solve for the unknown force or distance.

解杠杆系统时：确定支点，求顺时针力矩之和，求逆时针力矩之和，令两者相等，最后解出未知的力或距离。

4. Simple Machines & Mechanical Advantage | 简单机械与机械效益

Simple machines such as levers, pulleys and inclined planes allow a smaller effort to move a larger load. Mechanical advantage quantifies how much the machine multiplies the input force.

杠杆、滑轮和斜面等简单机械能让较小的动力移动较大的负载。机械效益用于量化机械将输入力放大了多少倍。

MA = L / E

MA – mechanical advantage (dimensionless), L – load (N), E – effort (N).

MA – 机械效益（无单位）；L – 负载（N）；E – 动力（N）。

in the same time. For an ideal (frictionless) machine MA = VR; for real machines MA < VR.

速比（VR）比较相同时间内动力移动的距离与负载移动的距离。理想（无摩擦）机械中 MA = VR；实际机械中 MA < VR。

VR = dₑ / dₗ

dₑ – distance moved by effort (m), dₗ – distance moved by load (m).

dₑ – 动力移动距离（m）；dₗ – 负载移动距离（m）。

5. Efficiency | 效率

Efficiency is the ratio of useful work output to total work input, expressed as a percentage. No real machine is 100% efficient due to friction and heat losses. Understanding efficiency allows engineers to select drives and design low‑loss systems.

效率是有用输出功与总输入功之比，用百分数表示。由于摩擦和热损失，任何实际机械都无法达到100%的效率。理解效率能帮助工程师选择合适的传动装置并设计低损耗系统。

η = (Work output / Work input) × 100%

For simple machines, efficiency can also be calculated using mechanical advantage and velocity ratio:

对简单机械，效率也可通过机械效益和速比计算：

η = (MA / VR) × 100%

measured in joules (J). Power is the rate of doing work: P = W / t, measured in watts (W).

功定义为力乘以沿力方向移动的距离：W = F × d，单位为焦耳（J）。功率是做功的速率：P = W / t，单位为瓦特（W）。

6. Drive Systems & Gear Ratio | 传动系统与齿轮比

Transmission systems using gears, pulleys or chains alter speed, torque and direction. The gear ratio is crucial for matching a motor's output to the required load behaviour.

使用齿轮、带轮或链条的传动系统可改变转速、扭矩和方向。齿轮比对于使电动机的输出与所需的负载特性相匹配至关重要。

Gear Ratio = Nₒᵤₜ / Nᵢₙ = Tᵢₙ / Tₒᵤₜ

Here N is rotational speed (rpm) and T is the number of teeth on a gear. The subscripts 'in' and 'out' refer to the driver and driven gears respectively.

其中 N 为转速（rpm），T 为齿轮齿数。下标“in”和“out”分别指主动轮和从动轮。

For a compound gear train, the overall ratio is the product of the ratios of each pair.

对于复合齿轮系，总传动比为各对齿轮副传动比的乘积。

When a small gear drives a larger gear, speed decreases and torque increases; when a large gear drives a smaller one, speed increases and torque decreases.

小齿轮带动大齿轮时，转速降低、扭矩增大；大齿轮带动小齿轮时，转速升高、扭矩减小。

drives, the velocity ratio is given by the ratio of the pulley diameters: VR = Dₒᵤₜ / Dᵢₙ.

对于带传动，速比等于带轮直径之比：VR = Dₒᵤₜ / Dᵢₙ。

7. Ohm's Law & Basic Circuits | 欧姆定律与基本电路

Ohm's Law relates voltage, current and resistance in a metallic conductor at constant temperature. It is the starting point for virtually all circuit analysis in the OCR Engineering specification.

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欧姆定律描述了恒温下金属导体中电压、电流和电阻之间的关系，是OCR工程考试中几乎所有电路分析的起点。

V = I × R

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

V – 电压（V）；I – 电流（A）；R – 电阻（Ω）。

Resistors in series add: R_total = R₁ + R₂ + R₃. Resistors in parallel combine as: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃.

串联电阻相加：R_总 = R₁ + R₂ + R₃。并联电阻的组合关系为：1/R_总 = 1/R₁ + 1/R₂ + 1/R₃。

Power dissipated in a resistor can be expressed in three useful forms, derived from Ohm's Law:

电阻的功率耗散可用由欧姆定律导出的三种实用形式表示：

P = V × I = I² × R = V² / R

onents. Always check the power rating of a resistor using P = I² × R to avoid overheating.

这些公式有助于计算能量损耗并选择合适的元件。请务必用 P = I² × R 核算电阻的额定功率，以免过热。

8. Potential Dividers & Sensing Circuits | 分压器与传感电路

A potential divider produces a fraction of the input voltage using two resistors in series. It is widely used with sensors – thermistors (temperature) and LDRs (light) – to create input signals for control systems.

分压器利用两个串联电阻产生输入电压的一部分。它通常与热敏电阻（温度）和光敏电阻（光照）等传感器结合使用，为控制系统提供输入信号。

Vₒᵤₜ = Vᵢₙ × (R₂ / (R₁ + R₂))

Vₒᵤₜ is the voltage across R₂ (the component you are measuring or using). When a sensor replaces R₂ and its resistance changes with the environment, Vₒᵤₜ changes accordingly.

Vₒᵤₜ为R₂两端的电压（待测量或使用的元件上的电压）。当用传感器替换R₂且其阻值随环境变化时，Vₒᵤₜ也会相应改变。

For a fixed R₁ and a variable R₂, the output voltage rises as R₂ increases. Designers place the sensor in the correct position to obtain either increasing or decreasing voltage with the measured quantity.

对于固定R₁和可变R₂，输出电压随R₂增大而升高。设计人员将传感器放置在合适的位置，以便使输出电压随被测量值升高或降低。

9. Density & Material Properties | 密度与材料性质

Density links the mass of a material to its volume and is vital when calculating component weight and material selection in design briefs. Several other measurable properties are used to compare engineering materials.

密度将材料质量与体积联系起来，在计算部件重量和根据设计任务书选材时至关重要。另外一些可测量的性质则用于比较工程材料。

ρ = m / V

ρ – density (kg/m³), m – mass (kg), V – volume (m³).

ρ – 密度（kg/m³）；m – 质量（kg）；V – 体积（m³）。

ductility are all key material properties you must be able to define and relate to stress‑strain graphs. The Young modulus, E, is the gradient of the linear portion of the stress‑strain curve: E = σ / ε.

硬度、刚度（杨氏模量）、韧性和延展性都是你需要定义并能与应力-应变图关联起来的关键材料性质。杨氏模量E是应力-应变曲线直线部分的斜率：E = σ / ε。

sity, then multiply by gravitational field strength g = 9.8 m/s² to find weight W = m × g.

在实际任务中，你可能需要先由部件体积和材料密度算出质量，再乘以重力场强度 g = 9.8 m/s² 求出重量 W = m × g。

10. Equilibrium & Free‑Body Diagrams | 平衡与受力图

A body is in static equilibrium when the net force and net moment acting on it are both zero. Free‑body diagrams are essential tools for visualising the forces and resolving them in X and Y directions.

当物体所受的合外力和合外力矩均为零时，它就处于静力平衡状态。受力图是直观展示力并将其沿X和Y方向分解的基本工具。

Σ Fₓ = 0   Σ Fᵧ = 0   Σ M = 0

Upward forces equal downward forces; leftward forces equal rightward forces; clockwise moments equal anticlockwise moments. This set of conditions allows you to solve for unknown reactions at supports.

向上的力等于向下的力；向左的力等于向右的力；顺时针力矩等于逆时针力矩。这一系列条件可用于求解支座的未知反力。

e horizontal, the horizontal component is F × cosθ and the vertical component is F × sinθ.

常利用三角函数分解力：若力F与水平方向夹角为θ，则水平分量为 F × cosθ，竖直分量为 F × sinθ。

11. Kinetic & Gravitational Potential Energy | 动能与重力势能

Energy analysis is required for systems involving movement and height changes. The OCR specification expects you to apply these formulas in contexts such as moving mechanisms and energy storage.

涉及运动和高度变化的系统需要进行能量分析。OCR考纲要求你在移动机构和储能等情境中应用这些公式。

KE = ½ × m × v²

KE – kinetic energy (J), m – mass (kg), v – velocity (m/s).

KE – 动能（J）；m – 质量（kg）；v – 速度（m/s）。

GPE = m × g × h

GPE – gravitational potential energy (J), g – gravitational field strength (N/kg), h – change in height (m).

GPE – 重力势能（J）；g – 重力场强度（N/kg）；h – 高度变化量（m）。

In an efficient system without air resistance or friction, the total mechanical energy (KE + GPE) is conserved. This principle helps determine speeds at different points on a roller‑coaster or in a pendulum.

在没有空气阻力或摩擦的理想系统中，总机械能（KE + GPE）守恒。这一原理可用于确定过山车或摆锤在不同位置的速度。

12. Unit Conversions & Useful Constants | 单位换算与常用常数

Engineering calculations require careful handling of units. A single conversion error can invalidate an entire solution. The table below lists the most common conversions and standard values used in Year 11 OCR Engineering.

工程计算需要严谨处理单位。一个换算错误就可能导致整个解答无效。下表列出了OCR 11年级工程中最常用的换算关系和标准值。

Quantity	Conversion / Value
Length	1 m = 1000 mm = 100 cm
Area	1 m² = 10³ × 10³ mm² = 10⁶ mm²
Force	1 N = 1 kg·m/s²
Pressure	1 Pa = 1 N/m²; 1 MPa = 10⁶ Pa
Energy	1 J = 1 N·m = 1 W·s
Power	1 W = 1 J/s; 1 kW = 1000 W
g (Earth)	9.8 N/kg ≈ 10 N/kg (for estimates)
Speed	1 m/s = 3.6 km/h

Always convert all given quantities to SI base units (metres, kilograms, seconds) before performing calculations. You should also memorise the prefixes: kilo (10³), mega (10⁶), centi (10⁻²) and milli (10⁻³).

计算前务必把所有已知量都转换为SI基本单位（米、千克、秒）。你还应熟记以下词头：kilo（10³）、mega（10⁶）、centi（10⁻²）、milli（10⁻³）。

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