📚 Formula & Theorem Quick Reference Handbook for Year 11 Eduqas Engineering | 11 年级 Eduqas 工程：公式定理速查手册
This quick reference handbook brings together the essential formulas and theorems you must master for the Year 11 Eduqas Engineering course. Use it for rapid revision, problem-solving, and to build confidence in applying core principles across mechanics, materials, and electronics.
本速查手册汇集了 11 年级 Eduqas 工程课程必须掌握的核心公式与定理。帮助你快速复习、解决问题，并熟练运用力学、材料和电子学中的基本原理。
1. Stress Formula | 应力公式
Stress (σ) measures the internal resistance of a material to an externally applied force. It is defined as the force acting per unit of cross‑sectional area.
σ = F / A
应力 (σ) 是材料抵抗外力的内部力量强度，定义为单位截面积上承受的力。
Stress is measured in pascals (Pa) in the SI system. 1 Pa equals 1 newton per square metre (N/m²). Engineers often use megapascals (MPa) for structural materials.
国际单位制中应力单位为帕斯卡 (Pa)，1 Pa = 1 N/m²。工程中常使用兆帕 (MPa) 表示材料强度。
2. Strain Formula | 应变公式
Strain (ε) quantifies the deformation of a material. It is the ratio of the change in length (ΔL) to the original length (L₀), making it a dimensionless quantity with no units.
ε = ΔL / L₀
应变 (ε) 描述材料的变形程度，是长度变化量 (ΔL) 与原始长度 (L₀) 的比值，因此没有单位。
Strain can be tensile (lengthening) or compressive (shortening). Since it is a ratio, strain is often expressed as a percentage when describing large deformations.
应变可分为拉伸应变和压缩应变。因为是比值，在描述较大变形时常以百分比表示。
3. Young’s Modulus | 杨氏模量
Young’s modulus (E) is a measure of a material’s stiffness. It is the constant of proportionality between stress and strain within the elastic limit, as described by Hooke’s law for materials.
E = σ / ε
杨氏模量 (E) 代表材料的刚度，是在弹性范围内应力与应变的比值，体现了材料遵循胡克定律的特性。
Alternatively, substituting the definitions gives E = (F L₀) / (A ΔL). Young’s modulus has the same units as stress, typically pascals or gigapascals (GPa) for metals.
也可写为 E = (F L₀) / (A ΔL)。杨氏模量的单位与应力相同，金属材料常用吉帕 (GPa) 表示。
4. Factor of Safety | 安全系数
The factor of safety (FoS) ensures that structures and components do not fail under normal working conditions. It is the ratio of the material’s ultimate stress (or failure stress) to the maximum allowable working stress.
FoS = Ultimate Stress / Working Stress
安全系数 (FoS) 确保结构或零件在正常使用条件下不会失效，是材料极限应力与许用工作应力的比值。
A factor of safety greater than 1 indicates a design margin. Typical values range from 1.5 for ductile materials under steady loads to much higher values for brittle materials or dynamic loading.
安全系数大于 1 表示有设计余量。韧性材料静载下的典型值约为 1.5，脆性材料或动载下则更高。
5. Hooke’s Law | 胡克定律
Hooke’s Law describes the linear elastic behaviour of springs and many engineering materials within the proportional limit. The extension (x) of an object is directly proportional to the applied force (F).
F = k × x
胡克定律描述了弹簧以及多种材料在比例极限内的弹性行为：物体的伸长量 (x) 与外力 (F) 成正比。
The spring constant k has units of N/m and indicates the stiffness of the spring. The law holds true until the elastic limit is exceeded; beyond that point permanent deformation occurs.
弹簧常数 k 的单位为 N/m，反映弹簧的刚度。该定律只在弹性极限内成立，超出后会产生永久变形。
6. Moments | 力矩
The moment of a force describes its turning effect about a pivot or fulcrum. It is calculated as the product of the force (F) and the perpendicular distance (d) from the pivot to the line of action of the force.
M = F × d
力矩描述力绕支点产生的转动效应，等于力 (F) 与力臂 (d，支点到力作用线的垂直距离) 的乘积。
Moments are measured in newton metres (N m). For a system to be in equilibrium, the sum of the clockwise moments must equal the sum of the anticlockwise moments.
力矩的单位是牛顿·米 (N m)。系统处于平衡时，顺时针力矩之和必须等于逆时针力矩之和。
7. Mechanical Advantage | 机械优势
Mechanical advantage (MA) is the ratio of the output force (load) to the input force (effort) in a machine. It indicates how much a mechanism multiplies the effort applied.
MA = Load / Effort
机械优势 (MA) 是机器的输出力 (负载) 与输入力 (施力) 的比值，表示机构将施加的力放大了多少倍。
For an ideal frictionless lever, MA can also be expressed as the ratio of the effort arm length to the load arm length. Real machines have friction, so actual MA is always less than the ideal value.
对于无摩擦的理想杠杆，MA 也可表示为施力臂长度与负载臂长度之比。实际机械存在摩擦，实际 MA 总小于理想值。
8. Velocity Ratio | 速度比
Velocity ratio (VR) compares the distance moved by the effort to the distance moved by the load in the same time. For gear systems, VR is determined by the tooth counts.
VR = Distance moved by effort / Distance moved by load
速度比 (VR) 是同一时间内施力端移动距离与负载端移动距离的比值。对于齿轮系统，VR 由齿数决定。
For gears: VR = Number of teeth on driven gear / Number of teeth on driver gear. For pulley systems, VR equals the number of rope segments supporting the load.
齿轮速度比公式：VR = 从动轮齿数 / 主动轮齿数。滑轮组的速度比等于承载负载的绳索段数。
9. Efficiency | 效率
Efficiency (η) is a measure of how successfully a machine converts energy input into useful work output. It is always less than 100% due to friction and other losses.
η = (Useful Work Output / Energy Input) × 100%
效率 (η) 表示机器将输入能量转化为有用功的程度，由于摩擦等损失，效率总小于 100% 。
Efficiency can also be calculated from mechanical advantage and velocity ratio: η = (MA / VR) × 100%. This makes it easy to assess performance when both ratios are known.
也可由机械优势与速度比计算效率：η = (MA / VR) × 100%。知道这两个比值即可快速评估机械性能。
10. Ohm’s Law | 欧姆定律
Ohm’s Law is the fundamental relationship in electrical circuits. It states that the current (I) flowing through a conductor is directly proportional to the potential difference (V) across it, provided temperature remains constant.
V = I × R
欧姆定律是电路的基本关系：在温度恒定时，流过导体的电流 (I) 与导体两端的电压 (V) 成正比。
Resistance (R) is measured in ohms (Ω). Rearranging gives I = V / R and R = V / I, which are equally important for circuit analysis.
电阻 (R) 的单位是欧姆 (Ω)。公式变形为 I = V / R 和 R = V / I，对电路分析同样重要。
11. Electrical Power | 电功率
Electrical power (P) is the rate of energy transfer in a circuit. It can be expressed in several forms by combining the basic definition with Ohm’s Law.
P = V × I = I² × R = V² / R
电功率 (P) 表示电路中能量转化的速率，结合欧姆定律可得到多种表达形式。
Power is measured in watts (W), where 1 W = 1 J/s. The choice of formula depends on the known quantities in the problem. P = I²R is useful for analysing heating effects in resistors.
功率单位为瓦特 (W)，1 W = 1 J/s。解题时根据已知量选择合适公式，P = I²R 特别适用于分析电阻的发热效应。
12. Resistors in Series & Parallel | 串联与并联电阻
When resistors are connected end‑to‑end, the total resistance is the sum of the individual resistances because the same current flows through each.
Rtotal = R₁ + R₂ + R₃ + …
电阻串联时，总电阻为各电阻之和，因为流过每个电阻的电流相同。
For resistors in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of the individual resistances. The overall resistance is always smaller than the smallest individual resistor.
1 / Rtotal = 1 / R₁ + 1 / R₂ + 1 / R₃ + …
电阻并联时，总电阻的倒数等于各电阻倒数之和，总电阻总是小于最小的单个电阻。
These rules allow engineers to design precise resistance values for sensors, voltage dividers, and current‑limiting circuits.
这些规则使工程师能够为传感器、分压电路和限流电路设计精确的电阻值。
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