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

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

This quick reference handbook gathers every essential formula and theorem needed for the Year 11 AQA Engineering course. It covers mechanics, electronics, materials and structural principles, all paired with clear explanations. Use it to reinforce your understanding and to speed up problem-solving in revision and exams.

本速查手册汇集了 Year 11 AQA 工程课程所需的全部核心公式和定理，涵盖力学、电子学、材料与结构原理，并配有清晰的解释。用它来巩固理解，在复习和考试中快速解题。

1. Stress and Strain | 应力与应变

Stress measures the internal force per unit area within a material when an external load is applied. Tensile stress acts to pull the material apart, while compressive stress squeezes it. It is calculated using the formula:

σ = F / A

where σ is stress in pascals (Pa), F is the applied force in newtons (N) and A is the original cross‑sectional area in square metres (m²).

应力衡量材料在外载荷作用下单位面积上的内力。拉伸应力将材料拉开，压缩应力则挤压材料。其计算公式为 σ = F / A，σ 为应力（帕斯卡），F 为施加的力（牛顿），A 为原始截面积（平方米）。

Strain is a dimensionless measure of deformation, comparing the change in length to the original length. The formula for strain is:

ε = ΔL / L₀

where ε is strain (no units), ΔL is the change in length (m) and L₀ is the original length (m). Strain can be tensile (positive) or compressive (negative).

应变是无量纲的变形量度，比较长度变化与原始长度。应变公式为 ε = ΔL / L₀，ε 为应变（无单位），ΔL 是长度变化（米），L₀ 是原始长度（米）。应变可以是拉伸（正）或压缩（负）。

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2. Young’s Modulus and Factor of Safety | 杨氏模量与安全系数

Young’s modulus describes the stiffness of a material in the linear elastic region. It is the ratio of stress to strain and is given by:

E = σ / ε

where E is Young’s modulus in pascals (Pa). A high value indicates a stiff material that deforms very little under load.

杨氏模量描述材料在线弹性区域的刚度，是应力与应变之比：E = σ / ε，E 为杨氏模量（帕斯卡）。数值高表示材料刚硬，在载荷下变形很小。

The factor of safety ensures that a structure is designed to withstand loads greater than the expected maximum. It is defined as:

Factor of Safety = Ultimate Stress / Allowable Stress

For example, if the ultimate tensile stress of a bolt is 800 MPa and the allowable working stress is 200 MPa, the factor of safety is 4.

安全系数确保结构设计能承受超过预期最大值的载荷。定义为：安全系数 = 极限应力 / 许用应力。例如，若螺栓的极限拉伸应力为 800 MPa，许用工作应力为 200 MPa，安全系数即为 4。

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3. Moments and Levers | 力矩与杠杆

A moment is the turning effect of a force about a pivot. It depends on the magnitude of the force and the perpendicular distance from the pivot to the line of action. The principle of moments states that for a system in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.

力矩是力对支点的转动效应，取决于力的大小和支点到力作用线的垂直距离。力矩原理指出，对于平衡系统，顺时针力矩总和等于逆时针力矩总和。

M = F × d

where M is the moment in newton‑metres (Nm), F is force in newtons (N), and d is the perpendicular distance from the pivot (m).

其中 M 为力矩（牛顿·米），F 为力（牛顿），d 为支点到力作用线的垂直距离（米）。

For a simple lever in equilibrium, the relationship is often written as:

F₁ × d₁ = F₂ × d₂

where F₁ is the effort, d₁ is the effort arm, F₂ is the load, and d₂ is the load arm.

对于平衡的简单杠杆，关系为 F₁ × d₁ = F₂ × d₂，其中 F₁ 为动力，d₁ 为动力臂，F₂ 为负载，d₂ 为负载臂。

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4. Work, Energy and Power | 功、能与功率

Work is done when a force moves its point of application. In mechanical systems, work transfers energy. The basic formula for work is:

W = F × d

where W is work in joules (J), F is the force in the direction of motion (N), and d is the distance moved (m).

力使其作用点移动时即做功。在机械系统中，功传递能量。功的基本公式为 W = F × d，W 为功（焦耳），F 为沿运动方向的力（牛顿），d 为移动距离（米）。

Gravitational potential energy is the energy stored due to height. It is calculated as:

GPE = m × g × h

s mass (kg), g is gravitational field strength (≈ 9.8 N/kg on Earth), and h is height (m).

重力势能是因高度而储存的能量，计算为 GPE = m × g × h，其中 m 为质量（千克），g 为重力场强（地球约 9.8 N/kg），h 为高度（米）。

Kinetic energy is the energy of motion and is given by:

KE = ½ × m × v²

where m is mass (kg) and v is velocity (m/s).

动能是运动的能量，公式为 KE = ½ × m × v²，m 为质量（千克），v 为速度（米/秒）。

Power is the rate at which work is done or energy is transferred. It can be expressed in two useful forms:

P = W / t

P = F × v

where P is power in watts (W), W is work (J), t is time (s), F is force (N), and v is velocity (m/s).

功率是做功或能量转移的速率。两种常用形式为 P = W / t 和 P = F × v，P 为功率（瓦特），W 为功（焦耳），t 为时间（秒），F 为力（牛顿），v 为速度（米/秒）。

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

Mechanical advantage (MA) describes how much a machine multiplies the input force. It is the ratio of load to effort:

MA = Load / Effort

Velocity ratio (VR) compares the distance moved by the effort to the distance moved by the load:

VR = Distance moved by effort / Distance moved by load

机械利益表示机械把输入力放大了多少倍，为负载与动力的比值：MA = 负载 / 动力。速度比是比较动力移动距离与负载移动距离：VR = 动力移动距离 / 负载移动距离。

Efficiency tells you how much of the input energy is converted to useful output. It can be expressed using work or power, and also in terms of MA and VR:

η = (Useful work output / Total energy input) × 100%

η = (MA / VR) × 100%

where η is efficiency (%). No machine can be 100% efficient due to friction and other losses.

效率表示有多少输入能量转化为有用输出。可用功或功率表示，也可用 MA 和 VR 计算：η = (有用功输出 / 总能量输入) × 100%，以及 η = (MA / VR) × 100%。η 为效率（%），由于摩擦等损失，没有机械能 100% 高效。

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6. Ohm’s Law and Electrical Power | 欧姆定律与电功率

Ohm’s law relates voltage, current and resistance in a circuit. It is fundamental to all circuit analysis:

V = I × R

where V is voltage in volts (V), I is current in amps (A), and R is resistance in ohms (Ω).

欧姆定律关联电路中的电压、电流和电阻，是所有电路分析的基础：V = I × R，V 为电压（伏特），I 为电流（安培），R 为电阻（欧姆）。

Electrical power is the rate at which energy is transferred in a circuit. It can be calculated using three equivalent forms:

P = I × V

P = I² × R

P = V² / R

where P is power in watts (W). Choose the form according to the known quantities.

电功率是电路中能量转移的速率，可用三种等效形式计算：P = I × V，P = I² × R，P = V² / R，P 为功率（瓦特）。根据已知量选用合适的形式。

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7. Resistors in Series and Parallel | 串联与并联电阻

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

Rₜₒₜ = R₁ + R₂ + R₃ + …

电阻串联时，总电阻等于各电阻之和，流过每个电阻的电流相同：Rₜₒₜ = R₁ + R₂ + R₃ + …

For parallel resistors, the total resistance is lower than the smallest individual resistor. The current divides between the branches. The general formula is:

1 / Rₜₒₜ = 1 / R₁ + 1 / R₂ + 1 / R₃ + …

For two resistors in parallel, a simplified formula is often used:

Rₜₒₜ = (R₁ × R₂) / (R₁ + R₂)

并联电阻时，总电阻小于最小的单个电阻，电流在支路中分配。通用公式为 1 / Rₜₒₜ = 1 / R₁ + 1 / R₂ + …，两个并联时常用简化形式 Rₜₒₜ = (R₁ × R₂) / (R₁ + R₂)。

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8. Potential Dividers and Sensors | 分压器与传感器

A potential divider uses two resistors in series to produce a fraction of the input voltage. The output voltage across the lower resistor is given by:

Vₒᵤₜ = Vᵢₙ × (R₂ / (R₁ + R₂))

where R₂ is the resistor across which the output is taken. This circuit is essential for using sensors such as LDRs and thermistors, where one resistor changes with light or temperature.

分压器用两个串联电阻产生输入电压的一部分。输出取自 R₂ 两端：Vₒᵤₜ = Vᵢₙ × (R₂ / (R₁ + R₂))。当

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